The calculation of the chemical diffusion coefficient for pure and mixed oxides with a simple defect structure. II. Numerical applications

Solid State Ionics 13 (1984) 21-31 North-Holland, Amsterdam

THE CALCULATION O F THE CHEMICAL D I F F U S I O N C O E F F I C I E N T F O R PURE AND MIXED OXIDES WITH A SIMPLE DEFECT STRUCTURE. II: NUMERICAL APPLICATIONS F. GESMUNDO Istituto di Chimica Fisiea Applicata dei Materiali del C.N.R., Lungobisagno 1stria 34, 16141 Genoa, Italy Received 9 August 1982 Revised manuscript received 17 February 1983

The expressions of the chemical diffusion coefficients for pure oxides and for ternary solid solutions of the (A,B)O type reported in part I of this paper are used for the calculation of these parameters for pure NiO and CoO and for their solid solutions at 1000*C. The results show that for both the pure oxides and their solutions the chemical diffusion coefficients measured from weight or from electrical conductivity changes differ according to models of the defect structure considering the simultaneous presence of metal vacancies with a different electric charge. However, the chemical diffusion coefficients for the solid solutions are more complex than those of a pure compound since they depend on the oxygen activity but also on the composition of the mixed oxide. Moreover it is shown that the chemical diffusion coefficients in these systems are, in general, functions also of ratio between the gradients of oxygen activity and oxide composition, a condition which leads to an anomalous type of behavior in one of the two fields of values of this ratio, either positive or negative, depending on the choice of the oxide BO.

1. Introduction In part I o f this paper [ 1] it has been shown that the general definitions o f the chemical diffusion coefficients for a pure oxide can be extended to include the solid solutions between two oxides having the same ideal value o f the metal/oxygen ratio. Equations for the calculation o f the two chemical diffusion coefficients for these systems have been derived under the assumption that the oxides form an ideal solution, contain only isolated metal vacancies and show a p-type conductivity. These expressions are used in the present paper to evaluate the chemical diffusion coefficients for pure NiO and CoO and for their solid solutions for which a reasonably accurate description o f the defect structure in terms o f metal vacancies o f different electric charge is available. In this way the behavior o f the calculated chemical diffusion coefficients for the mixed oxides as functions o f the oxygen activity and o f the oxide composition as well as o f the ratio between their gradients can be examined in details. As it is shown

later, the chemical diffusion coefficients considered here and defined b y analogy with the case o f a pure compound are different from the diffusion coefficient o f a solid solution obtained from the concentration profiles o f the two metal components in a diffusion couple and sometimes called "interdiffusion coefficient" [ 2 - 4 ] , which have already been measured for some solutions between oxides [2,3]. On the contrary, the chemical diffusion coefficients considered here have apparently never been measured for solid solutions o f the (A,B)O type, so that a comparison between calculated and measured values is not possible so far.

2. Calculations The calculation o f the chemical diffusion coefficients for the two pure oxides NiO and CoO at IO00oC has already been developed elsewhere [5] and the results are only briefly recalled here to facilitate the comparison with the results obtained for the solid so-

22

F. Gesmundo/Chemical diffusion coefficient for pure and mixed oxides. H

lutions between NiO and CoO. Both these oxides are p-type semiconductors containing metal vacancies as prevailing lattice defects [6] and show a negligible mobility of oxygen [6]. In addition they present a rather small concentration of lattice defects, so that the approximation of ignoring the interactions among defects is good for NiO and still reasonable for CoO [7,8]. For these oxides detailed models of the defect structure have been developed, considering the presence of all the three types of metal vacancies [8,9]. At IO00°C the constants of equilibrium for the formation of the different vacancies take the values K a = 5.20× 1 0 - 6 ; K b = 2 . 9 2 × l O - 3 ; K c = 5 . 3 3 X 10 -5 for NiO [9] while the corresponding constants for CoO are [8] Ka-= 1.37 X 10-3;Kb = 1.94X 10-2; K c = 1.89 X 10 -4, where K a , K b and K c are the equilibrium constants for the formation of the neutral, singly- and doubly-charged vacancies respectively, according to the reactions for the formation of the different defects reported in the previous paper [1 ]. In addition the equilibrium constant for the intrinsic ionizat i o n K d takes the value K d = 1.12 X 10 -16 for NiO [9] while for CoO it is in the range 1 0 - 8 - 1 0 -12 [10] and is taken here as K d = 10 -12 to avoid complications in the calculation o f ~ as pointed out elsewhere [5]. The equilibrium constants for the formation of NiO and CoO at IO00°C take the values [11] KNi O = 1.05 X 105 andK¢o o = 6 X 105 . Using these data, the concentrations of the different defects in the two pure oxides can be calculated as functions of the oxygen activity a o , as discussed in a previous paper [5 ]. The results are shown in fig. 1 (a) and (b) for NiO and CoO respectively. In both oxides the doubly-charged vacancies prevail under low oxygen activities and the singly-charged vacancies at high a O values but the crossing between the two concentrations occurs at higher oxygen activities for NiO (aO ~ 4 X 10 -3) than for CoO (a O -~ 5 × 1 0 ~ ) . In addition, in agreement with the values of the equilibrium constants for the formation of the vacancies, the concentrations of all the defects are smaller in NiO than in CoO under the same oxygen activity. The diffusion coefficients of the vacancies in pure NiO and CoO have been measured many times so far and the results have already been examined briefly previously [5]. The values adopted here are those used elsewhere for the calculation of the chemical diffusion coefficients in the two pure oxides [5], i.e.

-3

. . . . . . . . .

I=l 0 ---5

P

i .i I -11

. . -4. . . -3. . . . -2

-1

-

0

log il e -1

ol

o

-3

-5

-7

,./Iv,] "a-e

b)

-5

-~

'

-'a

'

-'a

'

-'1

'

0

log a o

Fig. 1. (a) Concentrations o f the different defects in pure NiO at 1000°C as functions o f the o x y g e n activity, (b) concentrations o f the different defects in pure CoO at IO00°C as functions o f the o x y g e n activity.

D v = 1.31 X 10 -7 cm 2 s - 1 for NiO [ 1 2 ] , a n d D v = 2.975 × 10 -7 cm 2 s -1 for CoO [8]. Use of these data and of the treatment presented earlier [1,5] allows the calculation of the chemical diffusion coefficients of the two oxides to be carried out: the results are shown in fig. 2(a) and (b). In both cases it is

F. Gesmundo/Chemical diffusion coefficient for pure and mixed oxides. II

4" I O " I ~ , N ~

. . . . . .

O,Oo-

3I

0

"

7

~

a) -

'

-~

'

-~

'

:~

'

-~

o

log ao

9.10"i 7 . 1 0 -~

5 " 1 0 -)

-6

b) -4

-2

log ao

0

Fig. 2. (a) Chemical diffusion coefficients of pure NiO at 1000*C as functions of the oxygen activity, (b) chemical diffusion coefficients of pure CoO at 1000°C as functions of the oxygen activity.

found that Do is greater than the corresponding/~, even though the difference is reduced, while both diffusion coefficients decrease with an increase o f a O . According to the defect models used for the two oxides and taking into account the fact that the diffusion coefficient of the vacancies has been assumed to be independent o f a O and the vacancy charge, the

23

change of/~ o and/~with a O appears to be related to the change in the nature of the predominating type of vacancy with a O. The experimental data are not in agreement with this prediction since they generally show no dependence of the chemical diffusion coefficient o n a O both for NiO [13-15] and CoO [13,16] while in two cases the chemical diffusion coefficient for CoO has been found to increase with a O [17,18]. This lack of agreement may be due to different reasons. In fact the values of D and/~o obtained from experimental measurements can be inaccurate to some extent due to a number of different complicating factors which have been discussed in details by Morin [19,20]. In the same way, a theoretical calculation of the chemical diffusion coefficients can also involve errors due to the approximations introduced in the procedure, such as those connected with the model used to represent the defect structure of the oxide considered, and in particular with the values of the equilibrium constants for the formation of the various defects, but also with the simplification of neglecting the interactions between defects. Another possible source of error is represented by the values of the diffusion coefficients of the defects, which are assumed to remain constant independently of the charge of the vacancies and of their overall concentration. In fact, a simple explanation of the increase with a O of the experimental values of/~ for CoO is that of assuming that the singly-charged vacancies diffuse faster than the doubly-charged, as has been proposed recently [18]. However, the most general condition is that the vacancy diffusion coefficient changes with a o even for the same kind of vacancy, as a result of the change in concentration of the defects which can affect the degree of interaction among them. In addition, the values of the diffusion coefficient of the defects are not known to a good accuracy. In fact, they can be obtained by using the self-diffusion coefficient of the metal in the oxide D M and the overall concentration of the vacancies 8 according to the relationship DM = DV~, where D V is assumed to be independent of the vacancy charge [5]. Thus, the inaccuracies in the experimental values o f D M and 6 result in corresponding uncertainties in D V . This can be seen more clearly by using the case of NiO as an example. In spite of the fact that this oxide has been extensively studied

24

F. Gesmundo /Chemical diffusion coefficient for pure and mixed oxides. H

there is still a large scatter in the data for both DNi and 5 [21]. Using the appropriate extreme values o f the measured ranges for both these parameters at 1000°C, the diffusion coefficient of the vacancies is found to range between 1.19 X 10 -8 and 2.36 X 10 -7 cm 2 s -1 [21,22], with a large degree of uncertainty. In this particular case, the dispersion of data concerning DNi, with a ratio between maximum and minimum values of 68.5 at IO00°C and under 1 atm. 0 2 , is much larger than that in the values of 6, with a corresponding ratio o f 2,80 [21 ], so that the former uncertainty can be considered as the more important source of error in the calculated value o f the chemical diffusion coefficient for NiO. However, even for this oxide the constancy of the experimental values o f the chemical diffusion coefficient with respect to a O as opposed to the decrease o f the calculated values of D w i t h an increase in a O can be attributed to a difference between the diffusion coefficients o f the vacancies with different charges. Thus, as it has been already concluded elsewhere [5 ], it appears that a more detailed knowledge than presently available of the nature and the diffusivities of the defects as well as more accurate experimental determinations of the chemical diffusion coefficients are probably needed before a more conclusive comparison between the two sets of data can be made for the pure oxides, even for the apparently simplest cases. The calculation of the chemical diffusion coefficients for mixed oxides is developed here for the solutions of NiO with CoO since these compounds are completely miscible at high temperatures [23 ], forming a system with an ideal behavior [24], and show a defect structure o f the same type as that o f the two pure oxides [2,24]. In the following Ni is taken as A-type ion with a mole fraction 1 - ~ and Co as Btype ion with a mole fraction ~. The concentrations o f the different defects in the mixed oxides can be obtained for any set of values o f a O and ~ using tlae approach developed in the previous paper [1 ] and the equilibrium constants for the formation of the various defects in the two pure oxides reported earlier. The effect o f a o and of ~ on the defect structure o f the solid solutions is shown in fig. 3 (a) and (b) where the concentrations o f the different defects are plotted as functions o f the oxide composition for two different values o f a O . It is seen that under a fixed a O the concentrations of all the vacancies in-

-4

-6

-8

112

04

08

0.8

-2 O

-3

-4

-5

b) -(

02

0.4

OS

0B

Fig. 3. (a) Concentrations o f the different defects in solid solutions N i O - C o O at 1000°C as f u n c t i o n s o f the mole fraction o f CoO (~) under a constant o x y g e n activity (a O = 1 0 - 4 ) , (b) concentrations o f the different defects in solid solutions N i O - C o O at 1000°C as functions o f the mole fraction o f CoO (~) under a c o n s t a n t o x y g e n activity (a O = 1).

crease with ~ as a result of the fact that CoO has a greater concentration of defects than NiO. Moreover the doubly-charged vacancies predominate at low and the singly-charged vacancies at high a O values as in the pure oxides.

F. Gesmundo/Chemical diffusion coefficient for pure and mixed oxides. H

25

Table 1 Tracer diffusion coefficients of Ni and Co at 1000°C in aix in solid solutions NiO-CoO for three different oxide compositions (~ = mole fraction of CoO)

D~qi (cm 2 s-1) DCo (cm 2 s-1)

0

0.47

1

1.39 × 10 -12 4.58 X 10 -12

3.41 × 10-11 1.09 × 10 -~°

2.60 × 10-1° 1.28 X 10-9

The next point to be considered for the mixed oxides concerns the change o f the diffusion coefficient o f the vacancies with the oxide composition. According to the previous discussion [ 1 ], the calculation o f D V requires a knowledge o f the j u m p frequencies o f the two ions and o f the corresponding correlation factors. At 1300 and 1445°C it has been found that the ratio between the j u m p frequencies o f the two ions does not change by more than 10% in the whole composition range, so that it may be considered to be approximately constant [2,24]. The tracer diffusion coefficients o f Ni and Co in their solid solutions have apparently not been measured so far at 1000°C. However, it is possible to extrapolate down to 1000°C the results obtained at higher temperatures concerning the two pure oxides and a solution with = 0.47 since they have been reported in the form o f an Arrhenius-type equation. The results o f this calculation are summarized in table 1, which reports the tracer diffusion coefficients o f the two ions at 1000°C and under the oxygen activity o f air. From these data the correlation factors for the diffusion o f Ni and Co can be calculated b y means o f the equations [24]:

fA = l

0.218 -

-

1 - ~ + ~D~ ]D~

fB = 1 --

0.218 /j + (1 - ~)D~/D~

,

(1)

with the results reported in table 2. While it is known that the correlation factor o f Co in NiO is smaller t h a n f 0 [2] the value obtained at 1000°C ( 0 . 2 8 2 ) a p pears too low since extrapolation o f the data o f Chen et al. [25] to this temperature y i e l d s f c o = 0.459. The small value calculated above is probably a result o f the inaccuracy o f the extrapolated values o f the tracer diffusion coefficients used. The increase o f fNi with ~ is also in agreement with the results obtained at higher temperatures [2]. Introduction o f the correlation factors reported in table 2 and o f the overall concentrations o f defects in air in the expressions o f D ~ , D~ reported in the previous part [1 ] yields the values o f the products a21-'Ni and a 2 I ' c o which are reported again in table 2. The ratio between the j u m p frequencies o f the two ions is finally obtained as 9.14, 4.32 and 5.97 for ~ = 0, 0.047 and 1 respectively. It seems therefore that at 1000vC the ratio between the

Table 2 Correlation factors and jump frequencies of Ni and Co in solid solutions NiO-CoO at 1000*C for three different oxide compositions (~ = mole fraction of CoO) I~

O

0.47

1

fNi

0.782

0.890

0.955

fCo

0.282

0.657

0.782

a21"Ni (cm 2 s-1)

9.96 X 10 -9

3.73 X 10- s

3.35 X 10- s

a2I'Co (cm 2 s-1)

9.10 X 10 -a

1.61 X 10-7

2 X 10-7

26

17. Gesmundo/Chemical diffusion coefficient for pure and mixed oxides. H

jump frequencies of the two ions is not even approximately independent of ~ at variance with the results obtained at higher temperatures. However the data available do not allow to obtain a reasonably accurate dependence of the jump frequencies on ~ because they refer only to three compositions and have been obtained by extrapolation of data measured at different temperatures. In view of this we adopt the simplification of obtaining the ratio of the jump frequencies from the diffusion coefficients of the vacancies in the two pure oxides which becomes thus independent of ~. In addition the calculation is simplified further by assuming that the individual jump frequencies of the two ions in the mixed oxides have the same values as in the corresponding pure oxides. This approximation is considered acceptable since the influence of FNi o n the value o f D V is larger in Ni-rich solutions for which D V is close to the value of pure NiO and the effect of Fco is larger in Co-rich solutions where D V is close to the value of pure CoO, while for intermediate composition the small value assumed for rNi is partly compensated for by the large value taken for FCo. Using eq. (24) of the previous paper [ 1] the diffusion coefficients of the vacancies in the two pure oxides can be expressed simply as D v(NiO) = 2a2FNi,

Dv(CoO ) = 2a2Fco ,

(2)

since the respective correlation factors become identical to f0 for the pure oxides. Since the change of the lattice parameter from NiO to CoO is only of about 2% at room temperature [23,26] and so a is practically independent of ~, the ratio between the diffusion coefficients in the two oxides yields also the ratio between the corresponding jump frequencies. Use of the values o f D V for the two pure oxides reported earlier yields F C o / F N i = 2.27. In addition eq. (2) yields for pure NiO: a2 l-,Ni = 6.5 X 10 -8 cm 2 s - 1 and for pure CoO: a2Fco = 1.48 X 10 -7 cm 2 s -1 : both these values are smaller than the corresponding values obtained from the tracer diffusion coefficients as reported in table 2, but the difference is larger for Ni. Use of the previous approximations and of the constancy of a allows to express the diffusion coefficient of the vacancies in the solid solution in the form DV = 1 [Dv(NiO)fNi( 1 _ ~) +Dv(CoO)fco~] ' (3) t °

o.g

0.8

0.7

0.6

0.2

0.4

0.6

0.8

Fig. 4. Correlation factors for the diffusion o f nickel and cobalt in solid solutions N i O - C o O at 1000°C as functions o f the mole fraction o f CoO (~).

where fNi,fCo are calculated from eqs. (25a) and (25b) of the previous paper [1] with Fco/FNi = 2.27. Finally, the term connected with the difference between the diffusion coefficients of the two ions appearing in eq. (36) of the previous paper [1] can be expressed as

()

e ~~- ao = ~ 0 [ D v ( N i O ) f N i - D v ( C ° O ) f c ° ]

~" (4)

The correlation factors for the diffusion of Ni and Co in the solid solutions are shown in fig. 4 as functions of the oxide composition. Both correlation factors increase regularly with the concentration of cobalt but fNi is always larger than fCo as it has already been shown for these mixed oxides [2]. The dependence of the diffusion coefficient of the vacancies on the composition of the oxide is shown in fig. 5 : D v has been calculated according to eq. (3) and shows a regular increase with ~ from the value typical of NiO to that of CoO. The chemical diffusion coefficient of the mixed oxides D 1 measured in presence of a gradient of oxide

F. Gesmundo/Chemicaldiffusion coefficient for pure and mixed oxides. II

27

•O7. x

3

-6 0.2

0.4

(16

(18

Fig. 5. Diffusion coefficient of the vacancies in solid solutions NiO-CoO at 1000°C as a function of the mole fraction of CoO (D. composition under constant oxygen activity is shown in fig. 6 as a function of ~ for some values of the oxygen activity. Under a constant ao,/~1 as an increasing

,..

o

6

K

Fig. 6. Chemical diffusion coefficient/~1 fo~ solid solutions NiO-CoO at 1000*C as a function of the mole fraction of CoO (~) for different values of the oxygen activity. Curve (1): a 0 = 10-4; curve (2): a O = 10-2; curve (3): a O = 1.

-5

-4

-3

-2

-1

log

0

ao

Fig. 7. Chemical diffusion coefficient/~1 for solid solutions NiO-CoO at 1000°C as a function of the oxygen activity for different values of the mole fraction of CoO (~). Curve (1): ~ = 0.9 ; curve (2): ~ = 0.5; curve ( 3): ~ = 0.1. function of ~ with a nearly linear dependence. This type of behavior is connected mainly with the change o f D V with ~ as shown in fig. 5 since the numerical resuits show that the term (ap/aS)ao in eq. (36) of [1] decreases only slightly with an increase in ~ and the term in e, which is negative since DNi < Dco, decreases also in the same direction. In particular the additional term in e is rather small with respect to the first term of D1, ranging from 7 to 10% of it at a o = 1. Moreover fig. 6 indicates that, for a fixed value of ~,/~1 decreases when a 0 increases. This situation is seen more clearly in fig. 7 where D1 is plotted as a function of a O for some values of g. Actually in this case the value of ~ along each curve represents a central value in a restricted range used to evaluate/~1 for given oxygen activity. The change in the m i n i m u m value o f a O from one curve to another is related to the different thermodynamic stability of the two pure oxides. In fact, a solid solution with a mole fraction ~ of cobalt is stable with respect to the corresponding alloy phase only above a m i n i m u m value o f a O (a~) which is a function of ~ through the equation a~) = [Kcoo(1 - ~) +KNio~]/(KcooKNio),

(5)

F. Gesmundo/Chemical diffusion coefficient for pure and mixed oxides. 1I

28

5 1

X

~r~

D2

,

'

o

50

'

R -6

-5

-4

-3

-2

-1 log ao

0

Fig. 8. Chemical diffusion coefficient/~2 for solid solutions N i O - C o O at 1000°C as a f u n c t i o n o f the o x y g e n activity for different values o f the mole fraction o f CoO (~). Curve (1): ~ = 1;curve (2): ~ = 0.75; curve (3): ~ = 0.5; curve (4): = 0.25 ;curve (5): ~ = 0.

where KNiO, Kco O are the constants o f equilibrium for the formation of NiO and CoO respectively as referred earlier and where an ideal behavior has been assumed for the alloy and oxide phases. "~ae decrease o f / ~ 1 with an increase in a O is connected with the decrease o f (aplaS)ao since D v is constant in this case while the term in ~ increases slightly but has a small effect since it is much smaller than the first term in eq. (36) of part I o f this paper [1] (about 4% at ~ = 0.5).

The dependence o f the chemical diffusion coefficient measured in presence of a gradient o f oxygen activity under a constant oxide composition D 2 on the oxygen activity is shown in fig. 8 for some values of~. The effect o f a O o n D 2 is of the same type as observed for the pure oxides and appears to be related again to the change in the type o f prevailing vacancy with a O . Moreover fig. 8 shows that under a given oxygen activity D 2 is an increasing function of ~ as it has already been found for/~1" Even in this case this behavior is related mainly to the change o f D V with the oxide composition. Finally the behavior o f the most general diffusion

Fig. 9. General chemical diffusion c o e f f i c i e n t / ) for a solid solution N i O - C o O with ~ = 0.5 at a0 = 1 as a function o f the ratio R b e t w e e n t h e gradients o f ao and ~ (dashed and d o t t e d lines represent D I and D2 respectively).

coefficient/~ as defined in eq. (35) o f the previous paper [1 ], is shown in fig. 9 as a function of the ratio R between the gradients of the logarithm o f the oxygen activity and that o f the oxide composition (R = a In ao/~) for the conditionsa O = 1 and ~ = 0.5. Since represents here the concentration o f CoO, which has a greater concentration of defects than NiO under the same oxygen activity, the term (a/i/a~)ao in eq. (35) of [1] is positive in the same way as the term (as/ a In a o ) ~, since under a constant composition the concentrations of the vacancies are increasing functions o f a O. Thus, when R is positive both terms in the numerator and the denominator of eq. (39) of [1] are positive yielding a chemical diffusion coefficient /~ which changes gradually from D1 to D 2 as R increases from zero to infinity. A different situation arises instead when R is negative, i.e. when the gradients o f a O and ~ have an opposite sign. In fact in this case both the numerator and the denominator of eq. (39) o f [1] become first equal to zero and then negative as R decreases. In addition the value o f R which produces a zero value of the numerator (R 1) is different from that which leads to a zero value for the denominator (R2). In fact these two special values of R are given by

F. Gesmundo/Chemical diffusion coefficient for pure and mixed oxides. I1 a8

R1=-[('~Lo+(-~)ao ~V] xF( ap L\a---~--Taol~ \a ln~,----~l~J

=

(6) '

as it can be seen by putting j~/v = 0 in eq. (33) of the previous paper [1] and R2=

\a~]aol,

08

If '

(7)

as obtained by putting J ' = 0 in eq. (34) of [1]. As a consequence o f this situation, when R = R 1 one obtains j~v = O, but under the same condition the gradient o f ~"is different from zero. Thus, to produce a fluxJ"equal to zero the chemical diffusion coefficient must become zero. This can be seen more clearly by writing D in the form

B = -y~?'/(a~/ax)

(8)

as obtained by comparing ~'to j~v. The other critical condition corresponds to R = R 2 since in this case the gradient of ~"becomes equal to zero, but at the same time j~v is different from zero. Thus, to produce a finite flux o f vacancies in absence o f a gradient of b', the value o f D b e c o m e s extremely large. Moreover, since the denominator of eq. (39) of the previous paper [1 ] changes sign when R decreases going through R 2 while the numerator maintains its sign, the chemical diffusion coefficient presents a discontinuity at R = R 2 going from --~ to +~. For still smaller values of R the chemical diffusion coefficient decreases gradually from +~ to D 2 . Introduction of the different parameters for the case examined in eqs. (6) and (7) yields the values R 1 = - 1 4 . 0 1 and R 2 = - 1 4 . 8 3 , with R 1 ~ R2. It is pointed out that, according to eqs. (6) and (7), R 1 and R 2 are both negative for this system as a consequence o f the choice o f to represent the concentration of CoO. In fact, since CoO has a greater concentration o f defects than NiO, both (~6/aDao and (ap/a~)ao are positive. It is also interesting to notice that in the range between R 1 and R 2 the chemical diffusion coefficient becomes negative, a result which may be interpreted as follows. The flux o f vacancies j~,v in the oxide as given by eq. (33) o f the previous paper [1] can also be expressed in the form

a- jao a-T n,

29

(9)

where n is the numerator of eq. (39) of the previous paper I1 ]. Taking into account the fact that ( a 6 / ~ ) a o is positive, the sign ofJ~rv will be the same as that of ~/Ox when n < 1 but opposite when n > 1, the former condition corresponding to R > R 1 and the latter to R < R 1- In the same way the gradient o f ~" may be expressed as

aS -- c u

d,

(10)

where d is the denominator o f eq. (39) of the previous paper [1 ] : in this case the gradient of ~"will have the same sign as that o f ~ when d > 0 (or when R > R2) but opposite when d < 0 (or when R < R2). Therefore j~v will have a sign opposite to that o f ~c'/ax f o r R > R 1 o r R < R 2 but the same sign as a~/ax for R 2 < R < R 1 . Introduction of these conditions in the expression o f / ~ as given by eq. (8) shows that D w i l l be positive f o r R > R 1 and R < R 2 but negative in the range between R 1 and R 2 . This is a result of the fact that for these combinations o f the gradients o f a O and ~ the flux o f vacancies as given by j~v occurs in the direction o f increasing~, so that a negative value of D is required to make J equalto j~v. The previous analysis of the dependence of D on R for fixed values o f a O and ~ shows that the extension o f the definition o f D from pure oxides to solid solutions introduces a complication concerning the values o f the chemical diffusion coefficients in case of presence of gradients o f both a O and ~ for one o f the two fields o f values o f the ratio R between the two gradients (either positive or negative). This occurs when the sign of R is such that the two gradients have an opposite effect on the concentration of the vacancies, one of them producing an increase and the other a decrease in the same direction. As discussed earlier, this corresponds to negative values o f R in the present system as a consequence o f the choice o f ~ to represent the concentration of CoO in the mixed oxides, so that the concentration o f the vacancies is an increasing function o f both a O and ~. Thus, for negative R values the gradients o f a O and ~ produce fluxes of vacancies in opposite directions which partly compensate each other and can also lead to a net flux equal to zero. In this field o f R values to make )"equal to j~v, as required

F. Gesmundo/Chemical diffusion coefficient for pure and mixed oxides. H

30

by the definition of/~, the chemical diffusion coefficient can take values even very different from D V , which re~esents usually the correct order of magnitude for D. The anomalous behavior of D in this condition appears to be a consequence of the definition of the chemical diffusion coefficient for these systems and has no special physical meaning. In view o f this, it seems that use and measurement o f / ~ in the critical field of R values should be avoided. No complications arise instead when the gradients o f a O and act in the same direction on the diffusion o f the vacancies and particularly in case of presence o f only one gradient, Which represents a condition more likely to be used in the experimental measurements. Thus, in spite o f the difficulties and o f the limitations mentioned above, the chemical diffusion coefficient can be a meaningful parameter even for the mixed oxides. Finally, the chemical diffusion coefficient measured from changes in the electrical conductivity of the samples D o is shown in fig. 10 as a function o f a O for different values o f ~. It can be seen that under a fixed value of ~ the chemical diffusion coefficient is

a decreasing function o f a O while under a fixed value of the oxygen activity D o increases with ~: this behavior is similar to that observed for D2 as reported earlier and may be interpreted in the same way. As discussed in part I [1], the complications arising for the calculation of D connected with the difference between D1 and D2 do not appear in the calculation o f D a as a consequence of the fact that in the expression o f J h. a term depending on the difference between the diffusion coefficients o f the two ions has been neglected. Thus Da results to be a function of a o and ~ only but not of the ratio between their gradients. The results reported above concerning the values of the chemical diffusion coefficients for the system N i O - C o O at 1000°C cannot be compared with experimental data since no measurements of D or D o as defined here appear to have been carried out so far on this or on similar systems. In fact the chemical diffusion coefficients considered here are different from the so-called interdiffusion coefficient defined by Wagner [27], controlling the rate of mixing of two metal components of a binary alloy or also that o f two oxides in a solid solution in absence o f a gradient o f oxygen activity. In fact the interdiffusion coefficient takes the form [28] /3 = [ D ~ + O]~ (1 - ~ ) 1 0 S ,

(11)

where O is the thermodynamic factor and S a term arising from the vacancy flow effect: both these terms are either equal or close to one for oxides [2,3]. On the contrary, introduction of eq, (27) into eq. (36) of part I of this paper [1 ] shows that the'chemical diffusion coefficient measured from weight changes in absence of a gradient of oxygen activity defined here has the form ap

a~ -6

-5

-4

-3

-2

-1

0

log Fig. 10. Chemical diffusion coefficient/)o foz solid solutions N i O - C o O at 1000°C as a function of the oxygen activity for different values of the mole fraction of CoO ( 0 . Curve (1): ~ = 1; curve (2): ~ -- 0.75 ;curve (3): ~ -- 0.5; curve (4): ~ = 0.25 ;curve (5): Ii = 0.

a

(12)

and is thus different f r o m / ) as a result of the different definitions o f the two diffusion coefficients. In spite of the absence o f experimental data concerning the chemical diffusion coefficient for oxide solid solutions, it can be anticipated that deviations between experimental and theoretically calculated

F. Gesmundo/Chemical diffusion coefficient for pure and mixed oxides. H values would almost certainly occur even for these systems. In fact the possible source of error for the experimental values remain essentially the same as before while new factors likely to produce errors in the calculated values of D arise, in addition to those already considered for the pure oxides. These include for example the assumption that the solution has an ideal behavior, the nature of the dependence of the equilibrium constants for the formation of the various defects on the oxide composition as well as the effect of the same parameter on the diffusion coefficient of the defects, and in particular on the values of the elementary jump frequencies and of the correlation factors for the two ions, which are not measured directly and are not well known. In view of this situation, the difference between the experimental and calculated values of the chemical diffusion coefficient for oxide solid solutions could be even greater than that observed for pure oxides. However the data presently available do not allow to discuss in a significant way the effect that these various factors can have on the difference considered above.

3. Conclusions The calculation of the chemical diffusion coefficients for the solid solution between NiO and CoO at 1000°C developed above shows that these parameters depend on both the oxygen activity and the oxide composition in a simple way when the oxide is subjected to the action of a gradient of only one of these two variables. When both gradients are present instead, an anomalous behavior of the chemical diffusion coefficients is produced in one of the two fields of values of their ratio, either positive or negative, corresponding to a condition in which the two gradients have an opposite effect on the concentration of the lattice defects. This situation limits in part the usefulness of this parameter for the mixed oxides. However no complications arise for the definition o f the chemical diffusion coefficient when the gradients of the two variables produce fluxes of defects in the same direction or when only one gradient is present. For the particular case of a solid solution with a p-type conductivity, showing an ideal behavior and containing only metal vacancies as lattice defects, the calculation of can be carried out in a simple way on the basis of

31

the knowledge of the defect structure of the two oxides and of the diffusional parameters of the two ions in the pure oxides and in their solid solutions.

References [1] F. Gesmundo, Solid State Ionics 12 (1984) 79. [2] W.K. Chen and N.L. Peterson, J. Phys. Chem. Solids 34 (1973) 1093. [3] G.J. Yurek and H. Schmalzried, Ber. Bunsenges. Physik. Chem. 78 (1974) 1379. [4] G.J. Yurek and H. Schmalzried, Bet. Bunsenges. Physik. Chem. 79 (1975) 255. [51 F. Gesmundo and F. Viani, J. Phys. Chem. Solids 42 (1981) 777. [6] P. Kofstad, Nonstoichiometry, diffusion and electrical conductivity in binary metal oxides (Wiley-Interscience, New York, 1972) p. 238. [7] C. Wagner, in: Progress in solid state chemistry, eds. J.O. McCaldin and G. Somorjai, Vol. 10 (Pergamon Press, Oxford, 1976) p. 3. [8] R. Dieckmann, Z. Physik. Chem. (NF) 107 (1977) 189. [9] C.M. Osburn and R.W. Vest, J. Phys. Chem. Solids 32 (1971) 1343. [ 10] B. Fisher and J.B. Wagner Jr., J. Appl. Phys. 38 (1967) 3838. [ 11 ] J.P. Coughlin, Bureau of Mines Bull. No. 542 (US Govt. Printing Office, Washington, 1954). [12] G.J. Koel and P.J. Gellings, Oxid. Met. 5 (1972) 185. [ 13] J.B. Price and J.B. Wagner Jr., Z. Physik. Chem. (NF) 49 (1966) 257. [14] J. Nowotny and R. Sadowski, J. Am. Ceram. Soc. 62 (1969) 24. [ 15 ] J. Nowotny and J.B. Wagner Jr., J. Am. Ceram. Soc. 56 (1973) 397. [16] J.M. Wimmer, R.N. Blumental and I. Branski, J. Phys. Chem. Solids 36 (1975) 269. [ 17 ] F. Morin, Can. Met. Quart. 14 (1975) 105. [18] G. Petot-Ervas, O. Radii and B. Sossa, in: IVth Europhysical Topical Conf., Lattice Defects in Ionic Crystals (Dublin, 30th Aug.-3rd Sept., 1982). [ 19] F. Morin, Can. Met. Quart. 14 (1975) 97. [20] F. Morin, J. Electxochem. Soc. 128 (1981) 2439. [21 ] K.N. Stratford and G. Smith, Oxid. Met. 14 (1980) 119. [22] F. Gesmundo and F. Viani, Solid State lonics 6 (1982) 33. [23] J. Robin, Ann. Chim. (Paris) 10 (1955) 395. [24] R. Dieckmann and H. Schmalzried, Ber. Bunsenges. Physik. Chem. 79 (1975) 1108. [25] W.K. Chen and N.L. Peterson, J. Phys. Chem. Solids 33 (1972) 881. [26] W.D. Johnston, R.C. Miller and R. Mazelsky, J. Am. Chem. Soc. 63 (1959) 198. [27] C. Wagner, Acta Met. 17 (1969) 99. [28] J.R. Manning, Acta Met. 15 (1967) 817.